Problem: $\dfrac{ 3b + 5c }{ -5 } = \dfrac{ -5b - d }{ -7 }$ Solve for $b$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 3b + 5c }{ -{5} } = \dfrac{ -5b - d }{ -7 }$ $-{5} \cdot \dfrac{ 3b + 5c }{ -{5} } = -{5} \cdot \dfrac{ -5b - d }{ -7 }$ $3b + 5c = -{5} \cdot \dfrac { -5b - d }{ -7 }$ Multiply both sides by the right denominator. $3b + 5c = -5 \cdot \dfrac{ -5b - d }{ -{7} }$ $-{7} \cdot \left( 3b + 5c \right) = -{7} \cdot -5 \cdot \dfrac{ -5b - d }{ -{7} }$ $-{7} \cdot \left( 3b + 5c \right) = -5 \cdot \left( -5b - d \right)$ Distribute both sides $-{7} \cdot \left( 3b + 5c \right) = -{5} \cdot \left( -5b - d \right)$ $-{21}b - {35}c = {25}b + {5}d$ Combine $b$ terms on the left. $-{21b} - 35c = {25b} + 5d$ $-{46b} - 35c = 5d$ Move the $c$ term to the right. $-46b - {35c} = 5d$ $-46b = 5d + {35c}$ Isolate $b$ by dividing both sides by its coefficient. $-{46}b = 5d + 35c$ $b = \dfrac{ 5d + 35c }{ -{46} }$ Swap signs so the denominator isn't negative. $b = \dfrac{ -{5}d - {35}c }{ {46} }$